Let $M$ be a closed oriented three-manifold with a smooth Riemannian metric $g_0$. For every finite time horizon $T<\infty$ and every accuracy parameter $\varepsilon>0$, there are positive surgery parameters

\begin{align*} 0<\delta\le \delta_0(g_0,T,\varepsilon),\qquad h=h(g_0,T,\varepsilon,\delta),\qquad D=D(g_0,T,\varepsilon,\delta), \end{align*}

with $\delta$ used as the constant cutoff function $\delta(t)\equiv\delta$ on $[0,T]$, with $h$ the neck radius, and with $D h^{-2}$ the trigger curvature scale, such that Perelman's $\delta$-cutoff construction produces a Ricci flow with surgery from $(M,g_0)$ on $[0,T]$ unless extinction occurs earlier. The canonical-neighbourhood and surgery-selection parts of the construction choose disjoint strong $\delta$-necks at scalar curvature comparable to $h^{-2}$, each cap inserted is $\delta$-close after scaling to the fixed standard cap model, and the post-surgery flow satisfies the [Hamilton-Ivey pinching estimate](/theorems/6022), the curvature bounds at the cutoff scale, and the noncollapsing constants required for the next smooth interval.